Melt 1 tbsp butter in a large non-stick skillet over med-high heat. Add the olive oil, onions, & salt. Cook until the onions are browned, about 10 minutes. Add the vinegar, turn off the heat, & transfer the onions to a small bowl. Do not wash the pan.
Mix 1 tbsp butter with the miso paste & mustard in a small bowl.
Spread two slices of bread with 1/2 tbsp butter each on one side and half of the miso butter each on the other side. Place the buttered side of each slice of bread in the skillet.
Divide the cheese & onion between each slice of bread on top of the miso paste.
Use the remaining butter to cover one side of the last 2 slices of bread and place them on top of the cheese & onion, buttered side out. Cook on med-low heat 4-5 minutes, until the bottom of the bread is golden brown. Do not press down with a spatula.
Flip the sandwiches over and brown the other side of the bread, 3-4 minutes more.
Serve with pickles - and a salad if you feel the need.